Sign and Symbol (e.g. inner product, outer product, etc) - SoojungHong/SomeMath GitHub Wiki

https://www.researchgate.net/figure/Symbols-and-notations-used_tbl1_320064840

What is inner product?

A dot product is a very specific inner product that works on ℝ𝑛 (or more generally 𝔽𝑛, where 𝔽

is a field) and refers to the inner product given by

(𝑣1,...,𝑣𝑛)⋅(𝑢1,...,𝑢𝑛)=𝑣1𝑢1+...+𝑣𝑛𝑢𝑛

More generally, an inner product is a function that takes in two vectors and gives a complex number, subject to some conditions.

In my experience, inner product is defined on vector spaces over a field 𝕂 (finite or infinite dimensional). Dot product refers specifically to the product of vectors in ℝ𝑛, however

Let...

𝑐∈ℂ

𝑢⃗ , 𝑣⃗ ∈ℝ𝑚×1 𝐴 , 𝐵∈ℂ𝕞×𝕟 Complex conjugate of 𝑐=𝑐⎯⎯⎯=𝑐∗ Absolute value of 𝑐=|𝑐|=𝑐⎯⎯⎯𝑐‾‾√

Now, according to wiki...

Dot product of 𝑢⃗ 

& 𝑣⃗ =𝑢⃗ ⋅𝑣⃗ =𝑢⃗ 𝑇𝑣⃗ . Inner product of 𝑢⃗ & 𝑣⃗ =⟨𝑢⃗ ,𝑣⃗ ⟩=𝑢⃗ 𝑇𝑣⃗

.

As mentioned in the chosen answer, the inner product is sometimes used as a generalization of the dot product (& the referenced article talks about this) but, in the interest of brevity/utility, I advise you to consider them equivalent unless you have reason to believe otherwise (i.e., strange notation, explicit definitions, context, etc.).

What is outer product?

In linear algebra, the outer product of two coordinate vectors is a matrix. If the two vectors have dimensions n and m, then their outer product is an n × m matrix. More generally, given two tensors (multidimensional arrays of numbers), their outer product is a tensor. The outer product of tensors is also referred to as their tensor product and can be used to define the tensor algebra.

The outer product contrasts with

the dot product, which takes as input a pair of coordinate vectors and produces a scalar.
the Kronecker product, which takes as input a pair of matrices and produces a matrix
and matrix multiplication.

Given two vectors

u = ( u 1 , u 2 , … , u m ) v = ( v 1 , v 2 , … , v n ) {\displaystyle {\begin{aligned}\mathbf {u} &=\left(u_{1},u_{2},\dots ,u_{m}\right)\\\mathbf {v} &=\left(v_{1},v_{2},\dots ,v_{n}\right)\end{aligned}}} {\displaystyle {\begin{aligned}\mathbf {u} &=\left(u_{1},u_{2},\dots ,u_{m}\right)\\\mathbf {v} &=\left(v_{1},v_{2},\dots ,v_{n}\right)\end{aligned}}}

their outer product u ⊗ v is defined as the m × n matrix A obtained by multiplying each element of u by each element of v:[1][2]

u ⊗ v = A = [ u 1 v 1 u 1 v 2 … u 1 v n u 2 v 1 u 2 v 2 … u 2 v n ⋮ ⋮ ⋱ ⋮ u m v 1 u m v 2 … u m v n ] {\displaystyle \mathbf {u} \otimes \mathbf {v} =\mathbf {A} ={\begin{bmatrix}u_{1}v_{1}&u_{1}v_{2}&\dots &u_{1}v_{n}\\u_{2}v_{1}&u_{2}v_{2}&\dots &u_{2}v_{n}\\\vdots &\vdots &\ddots &\vdots \\u_{m}v_{1}&u_{m}v_{2}&\dots &u_{m}v_{n}\end{bmatrix}}} {\displaystyle \mathbf {u} \otimes \mathbf {v} =\mathbf {A} ={\begin{bmatrix}u_{1}v_{1}&u_{1}v_{2}&\dots &u_{1}v_{n}\\u_{2}v_{1}&u_{2}v_{2}&\dots &u_{2}v_{n}\\\vdots &\vdots &\ddots &\vdots \\u_{m}v_{1}&u_{m}v_{2}&\dots &u_{m}v_{n}\end{bmatrix}}}

Or in index notation:

( u ⊗ v ) i j = u i v j {\displaystyle (\mathbf {u} \otimes \mathbf {v} )_{ij}=u_{i}v_{j}} {\displaystyle (\mathbf {u} \otimes \mathbf {v} )_{ij}=u_{i}v_{j}}

https://en.wikipedia.org/wiki/Outer_product

What is element-wise multiplication?

multiplication of elements from each vector that position in same place in the vector or matrix

In [1]: import numpy as np

In [2]: a = np.array([1,2,3,4])

In [3]: b = np.array([2,3,4,5])

In [4]: a * b Out[4]: array([ 2, 6, 12, 20])